The win means the Japanese will advance to the next phase of Asia's qualifying tournament for the 2022 World Cup if they defeat Myanmar when the teams meet in Japan on June 3.
Playing in an empty Fukuda Denshi Arena in Chiba after the Mongolians were forced to relinquish hosting rights due to quarantine regulations at home, the game often felt like little more than a training exercise for the Japanese.
Minamino put Japan ahead after 13 minutes before Osako claimed his first 10 minutes later, slotting home after neatly turning the Mongolian defence.
An own goal from Khash Erdene Tuyaa five minutes before the break capped a challenging first half for the Mongolians, and there was no letup in the second period.
Osako scored his second 10 minutes after the restart with another composed finish, and substitute Sho Inagaki hit Japan's seventh with 22 minutes remaining.
Japan coach Hajime Moriyasu praised his side's ruthlessness after they recorded their biggest win in World Cup qualifying.
"The players were focused on what they could accomplish rather than thinking about their opponent," Moriyasu said.
"In the middle of the game we changed our system, and I'm glad the players who came on for the first time did so well."